Trace class | EPFL Graph Search

Are you an EPFL student looking for a semester project?

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

In mathematics, specifically functional analysis, a trace-class operator is a linear operator for which a trace may be defined, such that the trace is a finite number independent of the choice of basis used to compute the trace. This trace of trace-class operators generalizes the trace of matrices studied in linear algebra. All trace-class operators are compact operators. In quantum mechanics, mixed states are described by density matrices, which are certain trace class operators. Trace-class operators are essentially the same as nuclear operators, though many authors reserve the term "trace-class operator" for the special case of nuclear operators on Hilbert spaces and use the term "nuclear operator" in more general topological vector spaces (such as Banach spaces). Note that the trace operator studied in partial differential equations is an unrelated concept. Suppose is a Hilbert space and a bounded linear operator on which is non-negative (I.e., semi—positive-definite) and self-adjoint. The trace of , denoted by is the sum of the serieswhere is an orthonormal basis of . The trace is a sum on non-negative reals and is therefore a non-negative real or infinity. It can be shown that the trace does not depend on the choice of orthonormal basis. For an arbitrary bounded linear operator on we define its absolute value, denoted by to be the positive square root of that is, is the unique bounded positive operator on such that The operator is said to be in the trace class if We denote the space of all trace class linear operators on H by (One can show that this is indeed a vector space.) If is in the trace class, we define the trace of bywhere is an arbitrary orthonormal basis of . It can be shown that this is an absolutely convergent series of complex numbers whose sum does not depend on the choice of orthonormal basis. When H is finite-dimensional, every operator is trace class and this definition of trace of T coincides with the definition of the trace of a matrix.

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related concepts (22)

Related courses (8)

Related lectures (104)

Quantum state

In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a quantum mechanical prediction for the system represented by the state. Knowledge of the quantum state together with the quantum mechanical rules for the system's evolution in time exhausts all that can be known about a quantum system. Quantum states may be defined in different ways for different kinds of systems or problems.

Trace class

Hilbert space

In mathematics, Hilbert spaces (named after David Hilbert) allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that induces a distance function for which the space is a complete metric space.

COM-406: Foundations of Data Science

We discuss a set of topics that are important for the understanding of modern data science but that are typically not taught in an introductory ML course. In particular we discuss fundamental ideas an

AR-202(o): Studio BA4 (Verschuere)

In the early days of the Anthropocene, the concepts of "Nature and Culture" are being revisited by a number of contemporary thinkers. The studio will take this paradigm shift as an opportunity to addr

EE-411: Fundamentals of inference and learning

This is an introductory course in the theory of statistics, inference, and machine learning, with an emphasis on theoretical understanding & practical exercises. The course will combine, and alternat

Signal Representations COM-406: Foundations of Data Science

Covers the norm of a matrix, operator, singular values, and unitary matrices in linear algebra.

Unsupervised learning: Young-Eckart-Mirsky theorem and intro to PCA EE-411: Fundamentals of inference and learning

Introduces the Young-Eckart-Mirsky theorem and PCA for unsupervised learning and data visualization.

Quantum Entanglement PHYS-758: Advanced Course on Quantum Communication

Delves into quantum entanglement, exploring entangled particles' state, evolution, and measurement.